Question

A study was done to look at the relationship between number of vacation days employees take each year and the number of sick days they take each year. The results of the survey are shown below.

Vacation Days	0	7	2	6	6	12	7	4	14
Sick Days	9	2	7	3	9	3	5	9	0

1. Find the correlation coefficient: r=r=   Round to 2 decimal places.
2. The null and alternative hypotheses for correlation are:
   H0:H0: ? ρ μ r  == 0
   H1:H1: ? μ r ρ   ≠≠ 0
   The p-value is:    (Round to four decimal places)
   
3. Use a level of significance of α=0.05α=0.05 to state the conclusion of the hypothesis test in the context of the study.
   • There is statistically significant evidence to conclude that there is a correlation between the number of vacation days taken and the number of sick days taken. Thus, the regression line is useful.
   • There is statistically significant evidence to conclude that an employee who takes more vacation days will take more sick days than an employee who takes fewer vacation days.
   • There is statistically significant evidence to conclude that an employee who takes more vacation days will take fewer sick days than an employee who takes fewer vacation days .
   • There is statistically insignificant evidence to conclude that there is a correlation between the number of vacation days taken and the number of sick days taken. Thus, the use of the regression line is not appropriate.
4. r2r2 =  (Round to two decimal places)
5. Interpret r2r2 :
   • 63% of all employees will take the average number of sick days.
   • There is a 63% chance that the regression line will be a good predictor for the number of sick days taken based on the number of vacation days taken.
   • There is a large variation in the number of sick days employees take, but if you only look at employees who take a fixed number of vacation days, this variation on average is reduced by 63%.
   • Given any group with a fixed number of vacation days taken, 63% of all of those employees will take the predicted number of sick days.
6. The equation of the linear regression line is:   
   ˆyy^ =  + xx   (Please show your answers to two decimal places)
   
7. Use the model to predict the number of sick days taken for an employee who took 6 vacation days this year.
   Sick Days =  (Please round your answer to the nearest whole number.)
   
8. Interpret the slope of the regression line in the context of the question:
   • The slope has no practical meaning since a negative number cannot occur with vacation days and sick days.
   • For every additional vacation day taken, employees tend to take on average 0.61 fewer sick days.
   • As x goes up, y goes down.
   
   
9. Interpret the y-intercept in the context of the question:
   • If an employee takes no vacation days, then that employee will take 9 sick days.
   • The average number of sick days is predicted to be 9.
   • The best prediction for an employee who doesn't take any vacation days is that the employee will take 9 sick days.
   • The y-intercept has no practical meaning for this study.

Answer 1

Vacation Days	Sick Days
0	9
7	2
2	7
6	3
6	9
12	3
7	5
4	9
14	0
r	-0.7931623

r = -0.79

n = 9

t = -1.8946

p-value = 0.0108

if p-value < alpha, we reject the null hypothesis

if p-value > alpha, we fail to reject the null hypothesis

since p-value < alpha, we reject the null hypothesis

hence option A) is correct

There is statistically significant evidence to conclude that there is a correlation between the number of vacation days taken and the number of sick days taken. Thus, the regression line is useful.

R-squared (R²) is a statistical measure that represents the proportion of the variance for a dependent variable that's explained by an independent variable or variables in a regression model.

r^2 = (-0.79)^2 0.6241

hence

There is a large variation in the number of sick days employees take, but if you only look at employees who take a fixed number of vacation days, this variation on average is reduced by 63%.

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